Quantum mechanics of 4-derivative theories
Alberto Salvio
aand
Alessandro Strumia
ba Departamento de F´ısica Te´orica, Universidad Aut´onoma de Madrid
and Instituto de F´ısica Te´orica IFT-UAM/CSIC, Madrid, Spain
b Dipartimento di Fisica dell’Universit`a di Pisa and INFN, Italia
CERN, Theory Division, Geneva, Switzerland
Abstract
A renormalizable theory of gravity is obtained if the dimension-less 4-derivative kinetic term of the graviton, which classically suffers from negative unbounded energy, admits a sensible quantization. We find that a 4-derivative degree of free-dom involves a canonical coordinate with unusual time-inversion parity, and that a correspondingly unusual representation must be employed for the relative quan-tum operator. The resulting theory has positive energy eigenvalues, normalizable wave functions, unitary evolution in a negative-norm configuration space. We present a formalism for quantum mechanics with a generic norm.
Contents
1 Introduction 2
2 The Ostrogradski classical canonical formalism 3
3 Quantum mechanics with negative norm 6
4 4 derivatives want Dirac-Pauli 13
5 Path-integral quantization 16
6 Interactions, Quantum Field Theory, probability 19
1
Introduction
Newton invented classical mechanics putting two time derivatives in his equation F = m¨x, which corresponds to a kinetic energy with 2 time derivatives, m ˙x2/2. Later Ostrograd-ski proofed a no-go theorem: non-degenerate classical systems with more than two time derivatives contain arbitrarily negative energies and develop fatal run-away instabilities [1]. Classically, they do not make sense.
The discovery that nature is relativistic and quantum opened the quest for an extension of Newtonian gravity. A century ago Einstein and Hilbert found the classical theory of relativistic gravity. However, its quantum version is not renormalizable in 3+1 space-time dimensions. Sticking to the observed number of space-time dimensions, a renormalizable extension of general relativity is found by adding terms quadratic in the curvature tensor to the Einstein-Hilbert Lagrangian, such that the graviton acquires a 4-derivative kinetic term. Stelle pro-posed and dismissed this extension [2] (see also [3–9]).
Recently the Higgs mass hierarchy problem brought interest to dimension-less theories. In this context, gravitons must have dimension 0 (being a dimension-less metric) and thereby must have a 4-derivative kinetic term. If these theories could make sense at quantum level, despite the negative classical energy, a great deal would be gained: relativistic quantum gravity, plus hierarchies among dynamically generated mass scales [8], plus inflation [10,11]. Quantization can eliminate arbitrarily negative classical energies. The following example is well known: the classical relativistic spin 1/2 field is described by a spinor Ψ(x) with Dirac LagrangianL = ¯Ψ(i/∂ − m)Ψ containing one time derivative. Treating Ψ as a classical field (as initially proposed by Schr¨odinger), and inserting the plane-wave expansion
Ψ(x) =
Z d3p
(2π)3p2E p
[ap,sup,se−ip·x+ b†p,svp,seip·x] (1)
in the Hamiltonian one finds negative energies in half of the configurations space:1
H = Z d3p (2π)3 Ep[a † p,sap,s− bp,sb†p,s], Ep = p m2+ ~p2 (2)
This classical arbitrarily negative energy is avoided by quantization with anti-commutators, if the vacuum state is appropriately chosen. Indeed, the two-state solution to {b, b†} = 1 shows that one can switch annihilation with creation operators by choosing the vacuum to be the state with lower energy.
The spin 0 and spin 1 relativistic fields (described by dimension-1 fields with 2 deriva-tives) do not have this issue: the negative-frequency solutions to the Klein-Gordon equation correspond to Hamiltonians with positive energy.
The general lesson is that quantization depends on the number of time derivatives. 1Unlike in the case of the Hydrogen atom emphasized by Woodard [12], where the instability eliminated by
The goal of this study is describing if/how systems with 4 derivatives can be quantized obtaining a consistent theory, in particular of quantum gravity. We will find that a unique structure emerges, that again involves switching annihilation and creation operators.
This will bring us into the territory of negative norm quanta, avoided like a plague by serious theorists that call them ‘ghosts’, and explored only by notorious crackpots such as Dirac [13], Pauli [14], Heisenberg [15], Pais, Uhlenbeck [16], Lee, Wick [17], Cutkosky [18], Coleman [19], Feynman [20], Gross [21], Hawking [22], ’t Hooft [23] and others, also more recently [24–32]. These works sometimes contain bizarre and confusing statements and obsolete motivations, together with interesting ideas and ad-hoc prescriptions.
This paper is structured as follows. In section2we review the canonical Ostrogradski for-malism. In section3we present negative norm quantum mechanics, the negative norm har-monic oscillator (section3.3), and the associated negative-norm representation of a canonical coordinate (section 3.4), with unusual parity under time-inversion T . In section4 we recall that a 4-derivative degree of freedom q(t) is described by two canonical coordinates: q1 = q and q2 = ˙q. While q1is T -even as usual, q2 is T -odd: we argue that thereby it naturally follows the negative-norm representation. The resulting quantum theory is unitary: time evolution preserves the negative norm. The path-integral formulation is discussed in section5. In sec-tion 6 we discuss the interacting theory, outline the extension to quantum field theory, and discuss the issue of giving a sensible interpretation to negative norms, via a postulate that generalizes the Born rule. Conclusions are given in section7.
2
The Ostrogradski classical canonical formalism
Let us now introduce the main issues in the simplest relevant case. Our final goal will be 4-derivative gravity; however the graviton components can be Fourier expanded into modes with given momentum and 4 time derivatives, and at leading order in the perturbative ex-pansion one has decoupled harmonic oscillators. So, we start considering a single mode q(t), described by the Lagrangian
L = −q¨2 2 +(ω 2 1+ω 2 2) ˙ q2 2 −ω 2 1ω 2 2 q2 2 −V (q) = − 1 2q( d2 dt2+ω 2 1)( d2 dt2+ω 2 2)q −V (q)+ total derivatives. where V (q) is some interaction. We assume real ω1, ω2, because we are interested in ghosts (negative kinetic and potential energy), not in tachyonic instabilities (potential unstable with respect to the kinetic term). The − sign means that the ghost is the state with larger ω; we choose ω1 > ω2 and don’t explicitly discuss here the degenerate case ω1 = ω2.
Ostrogradski introduced an auxiliary coordinate q2that allows to describe the 4-derivative oscillator in canonical Hamiltonian form (see also ref. [16] for a review of this method):
q1 = q, p1 = δL δ ˙q1 = (ω12+ ω22) ˙q +...q , q2 = λ ˙q, p2 = δL δ ˙q2 = −q¨ λ, (3)
where for a generic variable x we have introduced the variational derivative δL δx = ∂L ∂x − d dt ∂L ∂ ˙x + d2 dt2 ∂L ∂ ¨x + · · · . (4)
While Ostrogradski assumed λ = 1, we introduced an arbitrary constant λ. The system in eq. (3) can be solved for q and its time derivatives,
q = q1, q =˙ q2 λ, q = −λp¨ 2, ... q = p1− ω12+ ω22 q2 λ, (5)
and the Hamiltonian turns out to be
H = 2 X i=1 piq˙i−L = p1q2 λ − λ2 2 p 2 2− ω2 1 + ω22 2λ2 q 2 2+ ω2 1ω22 2 q 2 1+ V (q1). (6)
In view of its first term, the classical Hamiltonian H has no minimal energy configuration: this is the essence of the Ostrogradski no-go classical theorem. Using the Poisson parentheses { , } one computes the Hamiltonian equations of motion:
˙ q1 = {q1, H} = ∂H ∂p1 = q2 λ, p˙1 = {p1, H} = − ∂H ∂q1 = −ω21ω22q1− V0(q1), ˙ q2 = {q2, H} = ∂H ∂p2 = −λ2p2, p˙2 = {p2, H} = − ∂H ∂q2 = −p1 λ + ω2 1+ ω22 λ2 q2. (7)
For any λ they imply the classical Lagrangian equation of motion. Setting V = 0, it is
(d 2 dt2 + ω 2 1)( d2 dt2 + ω 2 2)q = d4q dt4 + (ω 2 1 + ω 2 2) d2q dt2 + ω 2 1ω 2 2q = 0. (8)
The corresponding classical solution, for given initial conditions q0, ˙q0, ¨q0, ... q0 at t = 0, is q(t) = −ω 2 2q0+ ¨q0 ω2 1 − ω22 cos(ω1t) + ω2 1q0+ ¨q0 ω2 1− ω22 cos(ω2t) − ω2 2q˙0 + ... q0 ω1(ω12− ω22) sin(ω1t) + ω2 1q˙0+ ... q0 ω2(ω12− ω22) sin(ω2t). (9) This is a well behaved oscillator without run-away issues because the positive-energy and negative-energy components are decoupled. Run-away solutions appear when they interact through a generic interaction, such as a V 6= 0.
2.1
Quantizing the Ostrogradski Hamiltonian
The classical equation differs from the usual 2-derivative equation d(q + ip)/dt = iω(q + ip), so that, trying to quantize the theory, we do not define the usual annihilation operator ai ∝ qi+ipi. Rather, it is convenient to define the operators aias the coefficients of given frequency:
The a1, a2 can be expressed in terms of canonical Hamiltonian coordinates: a1 = λω1ω22q1− iω12q2 + ip1λ − ω1p2λ2 2λω1(ω22− ω21) , (11) a2 = λω2 1ω2q1− iω22q2 + ip1λ − ω2p2λ2 2λω2(ω12− ω22) . (12)
Using the canonical quantization [qi, pj] = iδij one finds the commutation relations for the ai:
[˜a1, ˜a †
1] = −1, [˜a2, ˜a †
2] = 1, all other commutators vanish. (13) having normalised ˜a1 =p2ω1(ω12− ω22)a1 and ˜a2 =p2ω2(ω12− ω22)a2. The Hamiltonian is
H = −ω1 ˜ a1˜a † 1+ ˜a † 1˜a1 2 + ω2 ˜ a2˜a † 2+ ˜a † 2˜a2 2 . (14)
The state 1 with higher frequency ω1 > ω2 is a ghost.
As better discussed later in section3.3, this system can be quantized in two different ways:
1. Positive norm, negative energy. One redefines ˜a01 = ˜a†1, such that it has the usual commutation [˜a01, ˜a0†1] = 1. The vacuum state |˜0i is defined as usual by ˜a01|˜0i = 0 and ˜
a2|˜0i = 0. By solving this condition as a differential equation for ψ˜0(q1, q2) = hq1, q2|˜0i with pi = −i∂/∂qi one obtains the ground state wave function:
ψ˜0(q1, q2) = exp − q 2 1ω1ω2+ q22/λ2 2 (ω1− ω2) + iq1 q2 λω1ω2 . (15)
2. Negative norm, positive energy. The vacuum is now defined as a1|0i = 0 and a2|0i = 0. Using pi = −i∂/∂qi one obtains the ground-state wave function
ψ0(q1, q2) ∝ exp −q2 1ω1ω2+ q22/λ2 2 (ω1+ ω2) − iq1 q2 λω1ω2 . (16)
If λ = 1 the situation is bad, as emphasized by [12]: the positive-norm quantization gives a normalizable wave-function ψ˜0 but negative energies; the negative-norm quantization gives a ground state wave function not normalizable in q2 = ˙q. Excited states have the same problem. However, as we will show in section4, consistency demands the negative-norm Dirac-Pauli representation of a canonical coordinate which roughly amounts to choosing an imaginary λ, e.g. λ = −i. One then obtains positive energy, negative norm, and a wave function ψ0(q1, q2) normalizable in q1 and q2 = −i ˙q. As we will now discuss, despite the strange i factor, ˙q = iq2 as well as the Ostrogradski Hamiltonian H = iq2p1+ · · · = ˙qp1+ · · · are self-adjoint, so that time evolution is unitary.
3
Quantum mechanics with negative norm
We here discuss quantum mechanics with negative norm from a general point of view. Neg-ative norm states require putting some minus sign here and there. It is convenient to be more general and consider a Hilbert-like space with generic, possibly negative, constant norm (called Krein space by mathematicians) and develop a basis-independent formalism. This will let us to clarify confusions, in particular about self-adjoint operators that are represented (in some basis) by non-hermitian matrices, allowing us to understand the unusual imaginary λ introduced in the previous section.
We follow the notations used in general relativity, rewriting the quantum state metric as hn|mi = ηnm and defining the inverse metric (η)nm ≡ (η−1)nm, the contro-variant ket |ni = ηnm|
mi such that hn|mi = ηnm and hn|mi = δmn = hn|mi. Summations over repeated indexes are implicit. As usual, bras denote complex conjugate of kets.
A generic state |ψi can be expanded in either the ‘covariant’ or the ‘controvariant’ basis:
ψn≡ hn|ψi, ψn≡ hn|ψi. (17)
Then
|ψi = ψn|
ni = ψn|ni. (18)
A generic linear operator A can be written as a matrix in 4 different ways:
Anm≡ hn|A|mi, Anm≡ hn|A|mi, Anm ≡ hn|A|mi, Anm ≡ hn|A|mi. (19)
Then
A = Anm|nihm| = Anm|nihm| = Anm|nihm| = Anm|nihm|. (20) The components of the matrices are related by Anm = ηnn0An
0
m0ηm 0m
which is an iso-spectral transformation: the eigenvalues do not change because the matrix A gets left-multiplied by ηand right-multiplied by its inverse.
The unity operator is represented by 1nm= ηnmand 1nm= ηnm and expanded as
1 = ηnm|nihm| = ηnm|nihm| = |nihn| = |nihn|. (21)
Operator multiplication becomes, in components, (AB)nm = Ann0ηn 0m0
Bm0m. Expectation
values are given by hψ|A|ψi/hψ|ψi.
The adjoint A† of an operator A is defined, as usual, as the operator such that |ψ0i = A|ψi implies hψ0| = hψ|A†. Thereby for generic matrix elements one has hψ
2|A†|ψ1i ≡ hψ1|A|ψ2i∗, and the relation for the components
(A†)nm = A∗mn i.e. (A †
)nm = Amn∗ i.e. (A†)nm = (Amn)∗. (22)
The covariant components of a adjoint operator A satisfy the usual condition: a self-adjoint operator is described by a hermitian matrix, Anm. The same result holds for the
contro-variant matrix Anm. The mixed components satisfy a different condition, where com-plex conjugation is supplemented by an iso-spectral transformation: Anm = (ηA∗Tη−1)nm.2
A self-adjoint operator, A† = A has real expectation values hψ|A|ψi/hψ|ψi, although the matrix Am
n that represents it can be anti-hermitian.
The mixed components directly enter into the eigenvector equation A|ψi = Aψ|ψi:
Anmψm = Anm0ηm 0m
ψm = Aψψn or Anmψm = ηnn
0
An0mψm = Aψψn (23)
where Aψ is the eigenvalue. Let us now consider a self-adjoint operator H (later it will be the Hamiltonian), with eigenstates |Eni and eigenvalues En. The identity
hEn|H|Emi = hEn|EmiEm = En∗hEn|Emi (24)
tells that H can have three different kinds of eigenstates:
+) orthogonal eigenstates hEn|Emi = 0 with real En and norm hEn|Eni = +1;
−) orthogonal eigenstates hEn|Emi = 0 with real En and norm hEn|Eni = −1;
0) pairs of complex conjugated eigenvalues, En = Em∗ with hEn|Emi 6= 0 and zero norm, hEn|Eni = 0.
In the classical analogue, the latter possibility corresponds to a ghost which is also a tachyon, which is a different kind of instability, to be avoided even in absence of ghosts.
It is often convenient to choose a basis of eigenstates of H: |ni = |Eni. The associated contro-variant states |ni then satisfy H|ni = E∗
n|ni. In this basis the space splits into two sectors: positive norm and negative norm, plus the possible pairs of zero-norm states. The two sectors experience a joint dynamics only if the initial state has a quantum entanglement among them.
3.1
Unitary time evolution
The evolution equation i∂t|ψi = H|ψi becomes
i∂ ∂tψn = Hnmη mm0ψ m0 or i ∂ ∂tψ n = ηnn0H n0mψm. (25)
The norm of any state |ψ(t)i is conserved by time evolution if H is self-adjoint:
i∂ ∂thψ
0
(t)|ψ(t)i = hψ0|H − H†|ψi = 0. (26)
2Here T denotes the matrix transpose; ∗ denotes complex conjugation; † denotes the adjoint operation, that
generalizes the usual hermitian conjugation and reduces to it in the positive norm case. We never use † to denote hermitian conjugation of a matrix.
A self-adjoint Hamiltonian H leads to unitary time evolution. The explicit solution can be written as |ψ(t)i = U (t)|ψ(0)i with U (t) = Te−iR H(t) dt, where T is the usual time-ordering. In components, ψn(t) = Unmψm(0) = Unm0ηm 0m ψm0(0) or ψn(t) = Unmψm(0) = ηnn 0 Un0mψm(0). (27)
Having written generic-metric quantum mechanics in an abstract formalism that resem-bles as much as possible the usual positive-norm formalism, let us now emphasize the key differences. For simplicity, let us consider a time-independent H. One can then expand U = e−iHt=P∞
n=0(−iHt) n/n!.
• Writing U in mixed components, Unm is the naive exponentiation of the matrix Hnm. However, the mixed components of a self-adjoint H do not form a hermitian matrix. Rather, the self-adjoint condition in eq. (22) dictates that they are hermitian up to an iso-spectral transformation.
• The covariant components of a self-adjoint H satisfy the usual Hermiticity H∗
nm = Hmn. However, the covariant components Unm are not given by the naive matrix exponentia-tion of Hnm. Rather, extra metric factors appear to covariantize the expansion:
Unm= ηnm+ ηnn0(−iHt)n 0m0 ηm0m+ 1 2ηnn0(−iHt) n0r0η r0s0(−iHt)s 0m0 ηm0m+ · · · (28)
Correspondingly, the unitarity condition U†U = 1 written in covariant components is Un∗0nηn
0m0
Um0m = ηnm, while in mixed components one gets the usual Uk∗
nUkm = δmn.
Practical computations often employ perturbation theory, which can now be easily gener-alized to generic norm. Decomposing H = H0 + V (t), the Interaction picture is related to the Schroedinger picture as AI = eiH0tAe−iH0t where A is any operator (including V ). Time evolution is given by UI(ti, tf) = T e−i Rtf ti dt VI(t) = 1 − i Z tf ti dt0VI(t0) − Z tf ti dt0 Z t0 ti dt00VI(t0)VI(t00) + · · · . (29)
The above explicit form of UI shows that the energy conserved by quantum evolution (up to the usual quantum uncertainty ∆t ∆E ≥ ¯h) are the eigenvalues of H. Let us consider for example a time-independent interaction V and an initial state and a final state which are energy eigenstates with eigenvalues Ei and Ef. Defining Vfi = hf|V |ii, at first order one has
|hf|U | ii|2 ' Z t 0 dt0 ei(Ef−Ei)t0Vf i 2 = 4|V f i|2 |Ef − Ei|2 sin2 (Ef − Ei)t 2 t→∞ ' 2πt|Vf i|2δ(Ef − Ei). (30) This means that energy conservation reads Ef = Ei, up to the usual quantum uncertainty 1/(tf − ti). Higher order corrections give the usual sum over intermediate quasi-on-shell states.
3.2
Example: the indefinite-norm two-state system
Let us consider a two-state system: |+i with positive unit norm, and |−i with negative unit norm. Without loss of generality, by redefining the relative phase of the two states and adding a constant overall energy, one can trivially write the most generic self-adjoint Hamiltonian as
H = 1 2 |+i |−i h+| ER −iEI h−| iEI ER = 1 2 |+i |−i h+| ER iEI h−| iEI −ER (31) having used |±i = ±|
±i. We see that the Hnm components are hermitian, unlike the Hnm components. The eigenvalues of H are E± = ±Ewith E = pER2 − EI2/2. The corresponding eigenstates are |E+i = r γ + 1 2 |+i − i r γ − 1 2 |−i, |E−i = i r γ − 1 2 |+i + r γ + 1 2 |−i. (32) where γ = 1/p1 − E2
I/ER2 is a ‘boost factor’ that substitutes the usual mixing angle.
• If EI < ER the eigenvalues of H are real, the orthogonal eigenvectors satisfy hE±|E±i =
±1, and tend to get closer to the ‘light-cone’ of zero-norm states as EI increases. The components of U = e−iHtoscillate in time:
U =
|+i |−i
h+| cos(Et) − iγ sin(Et) pγ2− 1 sin(Et) h−| pγ2− 1 sin(Et) cos(Et) + iγ sin(Et)
. (33)
The unusual feature is that |h±|U |±i|2 oscillates between 1 and γ2 ≥ 1.
• In the critical case, ER= EI, such that γ = ∞, the two eigenstates become degenerate with energy E = 0. The two eigenvectors also become degenerate, and parallel to the zero-norm state ∝ |+i + i|−i. The evolution operator is
U = |+i |−i h+| 1 − iERt/2 ERt/2 h−| ERt/2 1 + iERt/2 . (34)
This exemplifies a more general result: zero-norm eigenstates with complex eigenvalues appear when, increasing the interaction, a level crossing between a positive-norm and a negative-norm eigenstate takes place; the Hamiltonian becomes degenerate at the critical transition.
• If the interaction EI is strong enough, EI > ER, one has a pair of complex conju-gated eigenvalues, with zero-norm eigenvectors that satisfy hE+|E−i = 1 and describe
in analogy to tachyonic states present in positive-norm theories. In the extreme limit EI ERthe eigenvalues of H are ±iEI/2, and the time evolution operator is:
U = |+i |−i h+| cosh(EIt/2) sinh(EIt/2) h−| sinh(EIt/2) cosh(EIt/2) . (35)
This runaway happens whenever H has a pair of complex eigenvalues E+ = E−∗, as clear writing time evolution in terms of energy eigenstates, |ψ(t)i = ψE+e−iE+t|
E+i+ψ
E−e−iE−t|
E−i.
Both the norm of |ψ(t)i and the real expectation value of H are preserved by time evolution:
hψ(t)|H|ψ(t)i hψ(t)|ψ(t)i =
E+ψE+ψE−∗+c.c.
ψE+ψE−∗+c.c. . (36)
3.3
The negative-norm harmonic oscillator
We here study the concrete system that lies at the basis of perturbative Quantum Field The-ory: the harmonic oscillator. As discussed by Lee and Wick [17] it admits two inequivalent quantizations: positive norm, and indefinite norm.
Let us first recall the standard oscillator, described (up to irrelevant constants) by the Hamiltonian H = 1 2(p 2+ q2)with [q, p] = i. Defining a = q + ip√ 2 , a † = q − ip√ 2 (37)
one has [a, a†] = 1and H = (aa†+ a†a)/2.
Let us next consider a more general system described by the following Hamiltonian and commutation relations: H = sH a†a + aa† 2 , [a, a † ] = s. (38)
For s = sH = 1 this reduces to the usual oscillator. We now show that s = sH = −1 defines another consistent positive-energy theory. The symbol a† here indicates the adjoint of a, which generalizes the Hermitian conjugate to negative norm.
We again define the vacuum as a|0i = 0 and the excited states as |ni = a†|n−1i/ √
n = (a†)n|0i/√n!. Its inverse is a|ni = s
√
n|n−1i. The state metric is ηnm ≡ hm|ni = snδnm. The norm is determined by the dynamics, and odd states have negative norm for s = −1. The inverse metric is ηnm = s−nδ
nm and the contro-variant states are |ni = s−n|ni. In components one has: a = |0i |1i |2i |3i · · · h0| 0 s 0 0 · · · h1| 0 0 √2s2 0 · · · h2| 0 0 0 √3s3 · · · h3| 0 0 0 0 · · · .. . ... ... ... ... . .. = |0i |1i |2i |3i · · · h0| 0 1 0 0 · · · h1| 0 0 √2 0 · · · h2| 0 0 0 √3 · · · h3| 0 0 0 0 · · · .. . ... ... ... ... . .. (39)
a†= |0i |1i |2i |3i · · · h0| 0 0 0 0 · · · h1| s 0 0 0 · · · h2| 0 √2s2 0 0 · · · h3| 0 0 √3s3 0 · · · .. . ... ... ... ... . .. = s |0i |1i |2i |3i · · · h0| 0 0 0 0 · · · h1| 1 0 0 0 · · · h2| 0 √2 0 0 · · · h3| 0 0 √3 0 · · · .. . ... ... ... ... . .. (40)
In components the commutation relations read
[a, a†]nm = (a · η · a†− a†· η · a)nm= sn+1δnm= sηnm i.e. [a, a†] = s X
n
|nihn| = s1 (41)
and the Hamiltonian is:
Hnm= (n + 1 2)sHs n+1 δnm = Enηnm i.e. H = ∞ X n=0 En|nihn| (42)
where En = (n +12)ssH are the Hamiltonian eigenvalues, H|ni = En|ni. We see that positive-energy eigenvalues are obtained for s = sH = 1(the usual case with positive H and positive norm), but also for s = sH = −1(negative H and negative norm).
Concerning the negative-norm case, s = −1, notice that the harmonic oscillator does not predict tachyonic ghosts with zero norm. Furthermore the matrix elements anm are not the hermitian conjugates of (a†)
nm, such that the operators q = (a + a†)/ √
2and p = i(a†− a)/√2 are represented by matrices qnm and pnm which are not Hermitian. This is why various authors who look at these matrices improperly speak of ‘anti-Hermitian’ operators. Neverthe-less, q and p are self-adjoint operators. We will now find their coordinate representation.
3.4
The negative-norm coordinate representation
Starting from the harmonic oscillator, we now describe a more general representation of a pair of canonical coordinate variables q, p. Parity flips q → −q and p → −p. In the harmonic oscillator case, this means a → −a and a† → −a†: so eigenstates |
ni with even (odd) n are even (odd) under parity. In the negative norm quantization, states with odd n also have negative norm. Going to the coordinate wave-function representation (we use the notation x for the coordinate, that later will become field space), this means that the norm is
hψ0|ψi = Z
dx [ψeven0∗ (x)ψeven(x) − ψodd0∗ (x)ψodd(x)] = Z
dx ψ0∗(x)ψ(−x). (43)
The corresponding unit operator is 1 =R dx| − xihx|. Switching to the formalism appropriate for generic norm, one has the norm hx0|xi = δ(x + x0). Thereby the controvariant state is
|xi = |
−xi and it satisfies the usual hx
0
|xi = δ(x − x0). As already discussed around eq. (17), a state can be expanded as |ψi = R dx ψ(x)|xi = R dx ψ(x)|xi with ψ(x) = hx|ψi and ψ(x) = hx|ψi = ψ(
What is emerging from the harmonic oscillator computation is a more general structure: a coordinate space representation of a pair q, p of conjugated canonical variables that differs from the usual positive-norm representation
q|xi = x|xi, p|xi = +i d
dx|xi (44)
which implies hx|p|ψi = (−id/dx)ψ(x) so that it satisfies hx|[q, p]|ψi = ihx|ψi.
The negative-norm coordinate representation, originally discussed by Dirac [13] and Pauli [14], is
q|xi = ix|xi, p|xi = + d
dx|xi. (45)
Although q looks anti-hermitian, taking into account the extra i as well as the negative norm, these unusual features combine to form a self-adjoint q:
hx0|q†|xi = hx|q|x0i∗ = [ix0δ(x + x0)]∗ = ixδ(x + x0) = hx0|q|xi. (46)
This means that hψ|q|ψi = R dq ψ∗(−q)iq ψ(q) is real. A similar result holds for p. When acting on wave-functions one has hx|q|ψi = −ixψ(x) and hx|p|ψi = (+d/dx)ψ(x), giving the desired [q, p] = i commutator. Defining momentum eigenstates as p|pi = ip|pi one finds hq|pi = eipq/
√
2π, hp0|pi = δ(p + p0). The operator q acts as hq|q|pi = (−d/dp)hq|pi. One can
again define |pi = |
−pi such that 1 =R dp |pihp|.
The i factor that differentiates the usual representation from the Dirac-Pauli representa-tion has an impact on the time-inversion parity. As usual, a positive energy spectrum demands that the time inversion symmetry is anti-unitary. Then, in the Dirac-Pauli quantization q is naturally T -odd and p is naturally T -even (while the opposite holds in the usual quantization, unless T is defined adding ad-hoc extra signs). This will play a key role in section4.
We are now ready to come back to the harmonic oscillator. Inserting into the condi-tion hx|a|0i = 0 the standard positive-norm representacondi-tion such that a = (q + ip)/√2 = (x + s d/dx)/√2gives a differential equation which implies the ground-state wave function ψ0(x) ∝ e−sx
2/2
. This is normalizable for s = 1 (positive norm) and non-normalizable for s = −1, where s was defined in eq. (38). This problem was emphaized e.g. by Woodard [12] that thereby dismissed the negative norm quantization as purely formal. However, the problem arises because the positive-norm representation of q, p was used together with the negative-norm oscillator: the problem is just a manifestation of the inconsistency of the as-sumptions. Consistency demands that the negative norm harmonic oscillator must be accom-panied by the negative-norm Dirac-Pauli coordinate space representation of the self-adjoint q, p operators, eq. (45). Then, the condition hx|a|0i = 0 leads to a normalizable wave func-tion for the ground state ψ0 ∝ e−x
2/2
, as well as for the excited states. The Dirac-Pauli choice thereby provides a self-consistent description of the negative-norm oscillator. Furthermore, as discussed in the next section, in the 4-derivative case the Pauli-Dirac representation is demanded by simple considerations.
norm hx|q|ψi T-parity hx|p|ψi T-parity harmonic oscillator with E > 0
positive xψ(x) even −i dψ/dx odd ψ0(q) ∝ e−q
2/2
and H = +1 2(q
2+ p2)
indefinite −ixψ(x) odd dψ/dx even ψ0(q) ∝ e−q
2/2
and H = −1 2(q
2 + p2)
Table 1: Coordinate representations of a pair of canonical variables [q, p] = i, and the associated ground-state wave functions for the positive-energy harmonic oscillator.
4
4 derivatives want Dirac-Pauli
As discussed in the previous section, and as summarized in table1, quantum mechanics has two faces: a canonical coordinate can be represented
i) in the usual way with positive norm;
ii) in the Dirac-Pauli way, with negative norm, eq. (45).
As we now show, theories with 4-derivatives want this latter quantization choice (that, in the gravitational case, corresponds to a renormalizable theory with positive energy).
A single 4-derivative real coordinate q(t) contains two degrees of freedom. The Ostro-gradski procedure (section2) rewrites the theory as a Hamiltonian system of two canonical coordinates, q1 = q and q2 = λ ˙q. The key new feature that arises in 4-derivative theories is that ˙q becomes an extra canonical coordinate. In the classical theory q2 is just an auxiliary variable, and λ is an irrelevant constant: Ostrogradski used λ = 1.
In the quantum theory, q1 and q2 are operators that allow to define the basis |q1, q2i. We now show that the usual quantization must be used for q1 and that the Dirac-Pauli quanti-zation must be chosen for q2, which is equivalent to (and more transparent than) fixing an imaginary λ and using the canonical representation.
As usual, the operator q1 = q is invariant under time-reversal t → −t, and thereby it can follow the usual T -even representation. On the other hand, the operator ˙q is T -odd, because of the time derivative: the time-inversion operator T transforms it as T ˙qT−1 = − ˙q. This is the novel key feature.
Taking into account that T is anti-unitary, one can equivalently define a usual T -even coordinate q2 = λ ˙q1 by choosing an imaginary λ.3 However, it is simpler to forget the λ factors and just declare that the self-adjoint operator ˙q is T -odd and thereby it follows the 3Alternative routes lead to the same conclusion. For example, one can use the T -even ¨q (instead of ˙q)
as second canonical coordinate. In the Ostrogradski formalism ¨q = −p2 is a momentum. So again one gets
a canonical coordinate with unusual T -parity (normally a momentum is T -odd). In general, the invariance of the commutation relation [q2, p2] = i under the anti-unitary time-inversion implies that q2 and p2 have
opposite T -parities. One can switch q2 ↔ p2in order to restore their usual T -parities: but their commutator
changes sign, implying again negative norm quantization. This is indeed what happens in the auxiliary variable formalism, used in various forms in the literature as an alternative to the canonical Ostrogradski formalism (see e.g. [22,24,29]). This formalism is convenient when dealing with quantum field theory instead of quantum mechanics with a finite number of degrees of freedom. In order to facilitate the contact, we summarize the
T-odd Pauli-Dirac representation. Then, the Ostrogradski Hamiltonian of eq. (6) is T -even. The states satisfy T |q, ˙qi = |q, ˙qi since ˙q has imaginary eigenvalues and since T is anti-unitary. The strange extra factor of i has been justified from first principles. A posteriori, it was not so strange. After all, it is well known that the self-adjoint spatial gradient is i~∇ rather than ~∇. In a relativistic theory, one could have guessed that similarly the self-adjoint time derivative is i∂/∂t rather than ∂/∂t. Loosely speaking, while from a classical perspective ˙q was the natural auxiliary variable, from a quantum perspective the natural extra coordinate operator is i ˙q.4
Using the Heisenberg representation, one has q(t) = U†(t)q(0)U (t) and ˙q = −i[q, H] = U†(t) ˙q(0)U (t)with unitary U , so q(t) keeps real eigenvalues and ˙q(t) keeps imaginary eigen-values at any t (these statements are not contradictory, given that q(t) also depends on p1(0) and p2(0)).
auxiliary variable formalism below. Restarting from the Lagrangian in eq. (3), we add zero as a perfect square containing an auxiliary variable a:
L = L + 1 2 ¨ q + (ω12+ ω22)q 2 − a 2 2 . (47)
Expanding the square cancels both the second-order and the fourth-order kinetic terms for q leaving L = −a¨q 2 + (ω 2 1− ω22)2 q2 8 − (ω 2 1+ ω22) aq 4 + a2 8 − V (q). (48)
Going to the free theory V = 0, we can diagonalize the kinetic and mass term through the field redefinition a =pω2 1− ω22(˜q2− ˜q1) q = (˜q2+ ˜q1)/pω21− ω22 i.e. q˜1,2= q 2 q ω2 1− ω22∓ a 2pω2 1− ω22 (49) obtaining two decoupled oscillators
L =q˙˜22− ω22q˜22 2 − ˙˜ q2 1− ω12q˜12 2 . (50)
From its classical solution, a = 2¨q + (ω2
1 + ω22)q, we see that a roughly corresponds to the Ostrogradski p2.
Furthermore, inserting such classical solution in eq. (49) one recovers the formalism used in [22] ˜ q2= ¨ q + ω12q pω2 1− ω 2 2 , q˜1= − ¨ q + ω22q pω2 1− ω 2 2 . (51)
4The possibility of converting a non-normalizable wave-function ψ 0 ∝ ez
2/2
into a normalizable one by restricting z = x + iy to the imaginary axis, rather than along the real axis, was presented as an ad hoc recipe to get something sensible in earlier works by Bender and Mannheim [25]. At the technical level, their approach differs from ours because they added an i factor to the variable q, rather than to ˙q. Our approach follows from general considerations, and has the advantage that q, p and thereby the Hamiltonian are self-adjoint (although their matrix representations look anti-hermitian), such that the generalization to an interacting theory will be immediate (section6.1).
The frequency eigenstates
We conclude this section by computing what the Dirac-Pauli representation adopted for q2 = ˙q implies for the frequency eigenstates. We restart from the Hamiltonian eq. (6) and bring it in diagonal form H = −1 2(˜p 2 1λ˜ 2 + ω12q˜ 2 1 ˜ λ2) + 1 2(˜p 2 2+ ω 2 2q˜ 2 2) (52)
through the canonical transformation
q1 = ˜ q2− ˜λ˜p1/ω1 pω2 1 − ω22 , q2 λ = ˜ p2− ω1q˜1/˜λ pω2 1− ω22 , p1 = ω1 ω1p˜2− ω22q˜1/˜λ pω2 1− ω22 , p2λ = ω22q˜2− ω1λ˜˜p1 pω2 1− ω22 . (53) which satisfies q1p1− q2p2 = ˜p2q˜2− ˜p1q˜1. For the sake of generality, we here allow for generic factors λ and ˜λ. The non-vanishing commutators, [˜q1, ˜p1] = iand [˜q2, ˜p2] = i, can be rewritten in terms of ˜a2 = pω2/2(˜q2 + i˜p2/ω2) and of ˜a1 = pω1/2(˜q1/˜λ − i˜λ˜p1/ω1) reproducing the Hamiltonian of eq. (14) and the commutators of eq. (13). The ground-state wave function is easily computed imposing h˜q1, ˜q2|˜a1,2|0i = 0 finding
ψ0(˜q1, ˜q2) ∝ exp − ω2 ˜ q2 2 2 + ω1 ˜ q2 1 2˜λ2 . (54)
For ˜λ = 1it is not normalizable [26]. It is normalizable if instead |Im ˜λ| > |Re ˜λ|.
The Dirac-Pauli representation for q2, p2 corresponds to imaginary λ. Imposing that q2, p1 are T -odd and that q1, p2 are T -even (i.e. that q2 and p2 have the unusual T parity) implies that ˜q1, ˜p2 are T -odd and that ˜q2, ˜p1 are T -even (i.e. that the canonical coordinates of the negative norm mode ˜q1 and ˜p1 have the unusual T parity). This is obtained for imaginary ˜λ.
As a check, let us connect the q1, q2 basis with the ˜q1, ˜q2 basis for generic λ and ˜λ. It is convenient to start from the T -odd basis ˜q1, ˜p2, in which the ground state wave function is
ψ0(˜q1, ˜p2) ∝ exp − p˜ 2 2 2ω2 + ω1 ˜ q2 1 2˜λ2 . (55)
Next, the transition to the T -odd variables p1, q2 is simply
hp1, q2|˜q1, ˜p2i ∝ δ ˜q1 ˜ λ − p1− q2ω21/λ ω1pω12− ω22 δ ˜ p2− p1− q2ω22/λ pω2 1 − ω22 . (56)
Inserting the change of variables dictated by the δ functions into ψ0(˜q1, ˜p2)one obtains
ψ0(p1, q2) ∝ exp − p 2 1+ 2ω1ω2p1q2/λ − ω1ω2(ω12+ ω1ω2+ ω22)(q2/λ)2 2ω1ω2(ω1+ ω2) (57)
where q2and p1are both complex and linked by Re p1 = ω21Re (q2/λ)and Im p1 = −ω22Im (q2/λ). ψ0 can be trivially analytically continued to real p1, q2. For λ = ±i it remains a bounded Gaus-sian. Finally, one performs the Fourier transform from p1 to q1, obtaining from ψ0(p1, q2)the ground state wave function ψ0(q1, q2), which agrees with eq. (16). The same equality holds for excited states, that can be computed acting with creation operators on the ground state.
5
Path-integral quantization
We now present the path-integral quantization of the same 4-derivative theory.
5.1
Path-integral for generic norm
Our generic-norm formalism makes easy to write down the equivalent path-integral formal-ism, an issue already considered in [21]. Inserting 1 = R dq |qih
q| at intermediate times tm = ti+ m dt one has hqf,tf| qi,tii = Y m Z dqmhqm+1,tm+1|qm,tmi. (58) Each step hqm+1,tm+1| qm,tmi can be evaluated as hqm+1|e−iHdt| qmi = Z dpmhqm+1|pmihpm|e −iHdt| qmi = Z dpm 2π e i[pm(qm+1−qm)−Hcldt] (59) having defined Hcl ≡ hp|H|qi hp|qi (60)
and used hq|pi = eipq/√2πand h
p|qi = e−ipq/ √
2π. The final result is the path integral
hqf,tf| qi,tii = Z Dq Dp eiR dt[p ˙q−Hcl] where Dq Dp = lim dt→0 Y m dqmdpm 2π (61)
and with boundary conditions q(ti) = qi, q(tf) = qf.
5.2
Path-integral for 4 derivative quantum theories
Applying the generic path-integral of eq. (61) to the 4-derivative oscillator in the canonical Ostrogradski formalism, one gets the transition amplitude
hq1f,q2f,tf| q1i,q2i,tii ∝ Z Dq1Dp1Dq2Dp2 exp i Z dt [p1q˙1+ p2q˙2− Hcl+ J1q1+ J2q2] (62)
where for generality we added currents J1,2(such that acting with functional derivatives with respect to them one can form more general matrix elements of time-ordered operators; J1 is T-even and J2 is T -odd). The Pauli-Dirac representation for ˙q manifests in two ways:
1) A propagator with an unusual − in its external state.
Rewriting the transition amplitude in the usual positive-norm formalism, it acquires an usual − sign, becoming hqf, − ˙qf, tf|qi, ˙qi, tii. In the limit tf → tione has hqf, ˙qf|qi, ˙qii = δ(qf− qi)δ( ˙qf−
˙
qi), so that the unusual − sign is equivalent to the Dirac-Pauli negative norm.5 Furthermore, the T -odd nature of ˙q is hardwired in the path-integral, as a geometrical feature. For each trajectory q(t) with boundary conditions q(ti,f) = qi,f and ˙q(ti,f) = ˙qi,f the time-inverted trajectory has the same action and the following boundary conditions:
qi → qf, qf → qi, q˙i → − ˙qf, q˙f → − ˙qi. (64)
Thereby the propagator given by the path-integral satisfies the identity
hqf, − ˙qf, tf|qi, ˙qi, tii = hqi, ˙qi, tf|qf, − ˙qf, tii (65)
which is equivalent to the operator identity hψf|ψii = hT ψi|T ψfi given that T |q, ˙q, ti = |q, ˙q, −ti.
2) An unusual classical Hamiltonian.
Inserting the Ostrogradski Hamiltonian of eq. (6) in the generic path integral of eq. (61) one gets the following classical Hamiltonian:
Hcl = hp1,p2|H|q1,q2i hp1,p2|q1,q2i = ip1q2+ p2 2 2 + ω2 1+ ω22 2 q 2 2 + ω2 1ω22 2 q 2 1 + V (q1). (66)
This is the same as eq. (6) with λ = −i. Hcl can be complex because q2, p2, in the Dirac-Pauli representation, have complex eigenvalues.6 Thanks to the unusual i, it is invariant under
time-reversal.
Let us now try to evaluate the path-integral. As usual, one can perform the Gaussian Dp1Dp2 path integrals. The Dp1 path-integral formally gives the Dirac delta function δ(q2 − λ ˙q1), allowing to eliminate the Dq2 path-integral, leaving
hqf, ˙qf,tf| qi, ˙qi,tii ∝ Z Dq exp i Z dt [L (q) + J1q + J2λ ˙q] , (67)
where L coincides with the original 4-derivative Lagrangian. By partial integration, the source term for ˙q can be transformed into a source for q or for ¨q (like in the auxiliary-field method). This computation however has three problems:
1. the Dp1 path-integral is, in general, divergent. Thereby the subsequent result is only formal.
5In the limit dt = t
f− ti→ 0 the classical action becomes
Scl' 6 dt3 qf− qi− dt ˙ qi+ ˙qf 2 2 +( ˙qf− ˙qi) 2 2 dt + · · · (63)
which is minimal for a motion with constant speed ( ˙qi + ˙qf)/2. The classical action satisfies
S(qf, ˙qf, tf; qi, ˙qi, ti) = −S(qi, ˙qi, ti; qf, ˙qf, tf)as well as Scl(qf, ˙qf, tf; qi, ˙qi, ti) = −Scl(qf, − ˙qf, ti; qf, − ˙qi, ti). 6The classical Hamiltonian is real if one instead uses the equivalent oscillator basis ˜q
2. the δ(q2 − λ ˙q1) always vanishes if q1 and q2 are real. Thereby the Dq2 path-integral is only formal.
3. Once interactions are turned on, the Lagrangian admits classical run-away solutions, reflected in the path-integral.
Given that the theory is well defined in the operator formalism, somehow this path integral must have a sense.
5.3
Euclidean Path-integral for 4 derivative quantum theories
A sensible path-integral is found by restarting from eq. (62) and continuing it to Euclidean time, it = tE, such that dq/dt = i dq/dtE i.e. ˙q = iq0. One gets the Euclidean path integral
hq1f,q2f,tEf| q1i,q2i,tEii ∝ Z Dq1Dq2Dp1Dp2 exp Z dtE(ip1q10 + ip2q20 − Hcl+ J1q1+ J2q2) . (68)
Now the Dp1 integral is convergent and gives δ(q2− q10), such that the Dq2 path integral just fixes q2 = q01. Next, the remaining terms in Hcl are a sum of positive squares so all other integrals are convergent. Performing them one finds the Lagrangian Euclidean path-integral:
hqf,q0f,tEf| qi,q0i,tEii ∝ Z Dq exp − Z dtE[LE(q) + J1q + J2q0] (69)
where the classical Euclidean Lagrangian corresponding to eq. (3) is
LE = 1 2 d2q dt2 E 2 + ω 2 1 + ω22 2 dq dtE 2 +ω 2 1ω22 2 q 2+ V (q). (70)
Let us now check the result. The classical free solution is
q(tE) = a1e−ω1tE + a2e−ω2tE + b1eω1tE + b2eω2tE. (71) It already contains run-away exponentials, characteristic of any Euclidean theory. Interac-tions compatible with the positivity of the action lead to an equally good path-integral. By imposing the boundary conditions q = q0 = 0at tEi = −∞and evaluating the classical action, one finds the normalizable ground-state wave function
hq, q0, tE = 0|0, 0, tE = −∞i ∝ exp − q 2ω 1ω2+ q02 2 (ω1+ ω2) + qq 0 ω1ω2 . (72)
This agrees with the ground-state wave-function ψ0(q1, q2)in eq. (16), that was computed in the Dirac-Pauli formalism in Minkowski space, after identifying q = q1 and q0 = q2. In other words, q0 = dq/dt
E = −idq/dt coincides with q2, as computed for λ = −i. The novel feature introduced by 4-derivatives is that q0 must not be continued into an imaginary −i ˙q (which would give divergent wave functions), because it already describes the T -odd variable q2,
which contains the Dirac-Pauli i factor of eq. (45).7 The final result is that the Minkowskian
theory is an unusual analytic continuation of the Euclidean theory.
6
Interactions, Quantum Field Theory, probability
Summarising, we so far considered a 4-derivative harmonic oscillator. One might think that we achieved nothing [26]. After all, a classical 4-derivative harmonic oscillator has no run-away problems, see eq. (9), given that it splits into two decoupled oscillators, one with neg-ative energy and one with positive energy. The classical trouble starts when they interact. In this section we will explain that we have achieved instead something useful in an interacting quantum field theory.
6.1
Adding interactions
The quantum formalism was so far developed for the harmonic oscillator (which corresponds to the modes of a free 4-derivative quantum field theory), finding that the quantum theory has a positive energy spectrum and no run-away behaviours. Adding interactions, the quantum interacting inherits all these good properties, as long as interactions are perturbative and as long as the interacting Hamiltonian H remains self-adjoint.
The second issue was the main obstacle to past attempts of adding ad-hoc unusual i factors in order to make the quantum free theory consistent [25] (normalizable wave functions and unitary evolution with negative norm and positive energy eigenvalues): adding extra complex factors can render interactions complex, ruining the theory [26].
In our approach the only extra i factor arose from a principled reason: ˙q is a T -odd coordinate that follows the negative-norm Dirac-Pauli representation. This satisfies all the properties of quantum mechanics, as generalised to negative norms: ˙q itself is self-adjoint, like q and ¨q. Thereby any interaction which is a real function of them is self-adjoint. Our procedure immediately generalizes to the interacting case (in agravity [8] all interactions are dictated by general covariance).
The perturbativity assumption means that, as long as the energy spectrum of the free oscillator gets slightly distorted by interactions, the energy eigenvalues will remain real and bounded from below (strongly interacting theories could also be good; however they seem not needed for the physical application to agravity [8]).
One might worry that, even if all energy eigenvalues are positive, the theory possesses negative-norm states with hψ|H|ψi < 0. Eq. (29) shows how transition amplitudes can be computed trough perturbation theory: we see that the energy eigenvalues are the quantity 7Hawking and Hertog [22] found a non-normalizable Minkowskian wave-function because they expressed
eq. (72) in terms of ˙q, which is not the canonical coordinate q2 appropriate for 4-derivative theories. Their
proposal of integrating out ˙q in the Euclidean before performing the analytic continuation to the Minkowskian is not necessary: the Minkowskian wave functions are normalizable if the appropriate analytic continuation is performed.
that enters into conservation of energy. Thereby a theory where all eigenvalues of H (of H0 in the perturbative expansion) are positive is consistent. As usual, perturbative computations can be systematised in terms of the propagator. By expressing q = q1 in terms of the annihi-lation and creation operators ai, a
†
i through eq.s (11) and (12) and using the commutation relations [˜ai, ˜a
†
i] = si we find the propagator
h0|T q(t)q(t0)|0i = h0|θ(t − t0)q(t)q(t0) + θ(t0− t)q(t0)q(t)|0i (73a)
= 1 ω2 1 − ω22 X i si 2ωi [eiωi(t−t0)θ(t0− t) + eiωi(t0−t)θ(t − t0)] (73b) = i 1 ω2 1 − ω22 Z dE 2π X i sie−iE(t−t 0) E2− ω2 i + i (73c) = Z dE 2π −i e−iE(t−t0) (E2− ω2 1 + i)(E2− ω22+ i) . (73d)
where is a small positive quantity and we used s1 = −1and s2 = 1 in the last step.
One might worry that, using the Heisenberg picture, operators satisfy the time evolution equation ˙A = −i[A, H], which looks dangerously similar to the classical equation of motion, as given by Poisson parentheses, which has run-away solutions. However, the quantum so-lutions are equal to the classical soso-lutions only in a free theory. In general operators are not numbers, and the difference (in particular, the Pauli-Dirac representation) manifests when non-linear interactions are present. As well known, the Heisenberg equations are in general solved by A(t) = U†(t)A(0)U (t). So, all good properties of negative norm states found in the Schr¨odinger picture remain valid in the Heisenberg picture, given that they are equivalent.
6.2
Extension to quantum field theory
As well known, a single harmonic-oscillator degree of freedom q(t) is the building block for a field such as φ(t, x, y, z) or gµν(t, x, y, z). The expansion of a field in Fourier modes with given momentum works in the 4-derivative case similarly to the 2-derivative case. As long as, at the end, we are only interested in S-matrix elements, all the detailed structure of the quantum mechanical theory, such as the wave-functions, gets hidden behind the commutation relations of eq. (13), which hold separately for each mode. The usual i prescription for the field propagator dictates that amplitudes can be analytically continued from the Euclidean. Details will be presented elsewhere.
One would like to claim that quantum field theory inherits all good properties of quantum mechanics also when negative norms are present. However, while in quantum mechanics interactions can easily satisfy the condition that avoids ‘tachionic ghosts’ (namely, the inter-action strength between two opposite-norm states must be smaller than their energy differ-ence as discussed in section 3.2), any interesting quantum field theory leads to situations that might violate this condition. The simplest situation where this occurs is the decay of a ghost (for example a massive spin 2 graviton at rest), which can be degenerate with a multi-particle state (for example two photons going in opposite directions with energy equal to
half of the ghost mass), such that the interaction, no matter how small, can be smaller than the energy difference. Actually, the ghost is degenerate with an infinite number of similar states, such that an appropriate limit procedure is needed: in the positive norm case, en-tropic considerations allow to interpret this situation as particle decay. A 4-derivative kinetic term Π(p) = −(p2− m2
1)(p2− m22)acquires a positive imaginary part. We will explore if ‘ghost decay’ can be interpreted like in [24].
6.3
Ghost does not play dice?
So far we carefully avoided talking about probabilities.
The theory is unitary in a negative-norm space. Thereby the only remaining difficulty is assigning an interpretation to states that entangle positive norm components with negative norm components. The Copenhagen interpretation added an extra ingredient external to the deterministic formalism of quantum mechanics: the Born postulate, according to which:
“when an observable corresponding to a self-adjoint operator A is measured in a state |ψi, the result is an eigenvalue Anof A with probability
Pn =
hψ|Πn|ψi
hψ|ψi where Πn =
|nihn|
hn|ni (74)
is the projector over the eigenstate |ni of A”.
For positive norms, these Pn satisfy the probability rules 0 ≤ Pn ≤ 1 and PnPn = 1; the average value of A satisfies hψ|A|ψi/hψ|ψi =P
nAnPn.
At the moment we do not have a satisfactory generalisation to indefinite norm. Even worse, the Born postulate is unsatisfactory by itself, given that it describes a non-local collapse of the wave-function [33]. In order to make progress, one needs to operate close to the heart of quantum mechanics. As well known this presents fatal risks: physicists tend to become philosophers. We conclude by listing some interpretations of quantum mechanics, equivalent to the Copenhagen interpretation, which could lead to a satisfactory indefinitive norm quantum mechanics.
1. Feynman clarified the ontological basis of the Born postulate: it agrees with experi-ments, so ‘shut up and compute’. All experiments have so far been performed with positive norm states. The negative norm graviton predicted by agravity is beyond the reach of present experiments. On the one hand, this is good because it means that Einstein’s general relativity is recovered at large distances; on the other hand, however, we do not have experimental guidance. Lee and Wick proposed that the interpretation issue is bypassed, given that in quantum field theory we can only observe asymptotic states, which are made of positive-norm quanta [17]. The Lee-Wick idea may be applied to the gravitational theory proposed by Stelle [2], as discussed in [5,34].
2. Any self-adjoint Hamiltonian H gives unitary evolution with respect to many different norms, since each energy eigenstate evolves picking just a phase. Defining ghost parity G to be the metrics in the special basis of energy eigenstates and |ψi = G−1|
ψi, a possible generalization of the Born postulate to generic norm is (see also [25])
Pn = hψ|Πn|ψi where Πn = |nihn|. (75)
The example of section 3.2 gets converted into normal oscillations with mixing angle sin22θ = E2
I/ER2. However, hψ|A|ψi is real but does not have a probabilistic interpreta-tion, while hψ|A|
ψi has a probabilistic interpretation but can be complex.
3. Various authors claim that the Born postulate is just an emergent phenomenon (some-how like friction) that follows from the fundamental deterministic equations when ap-plied to complex systems in view of spontaneous decoherence [35].
4. Cramer [36] proposed a “transactional interpretation”, claiming that EPR non-locality results from a cancellation of advanced and retarded waves, in a time-symmetric set-up (see also [37]) inspired by the analogous formulation of classical electro-dynamics proposed by Dirac and Feynman-Wheeler. The hψ0|ψi amplitude in the Dirac-Pauli co-ordinate representation supports the interpretation as being the overlap of a wave ψ moving forward in time with a wave ψ0 moving backwards in time.
We plan to further investigate such issues.
7
Conclusions (so far)
We presented the quantization of 4-derivative theories, finding that a unique structure emerges. We can summarise it as follows.
Quantum mechanics has its usual visible face, where a coordinate operator q is repre-sented as q|xi = x|xi. But quantum mechanics also has a hidden face, where q|xi = ix|xi, as first pointed out by Dirac and Pauli. Both q and p of a canonical pair [q, p] = i are self-adjoint in both representations. The main difference is that the usual representation implies positive norms and q is naturally even under time reflection T , while the DP representation leads to states with indefinite norm and to a naturally T -odd q (in view of the i factor and of the fact that T is anti-unitary)
The Ostrogradski formulation of a 4-derivative degree of freedom q(t) (summarised in section 2) employs two canonical coordinates: q1 = q and q2 = ˙q. For the first time we have observed that q1, which is T -even, naturally follows the usual representation, while q2 which is T -odd, naturally follows the Dirac-Pauli negative-norm quantization. This leads to a sensible quantum theory with positive energies and normalizable wave-functions, as discussed in section4.
In section 3 we presented a new formalism appropriate for generic-norm quantum me-chanics, introducing ‘covariant’ |ni and ‘contro-variant’ |ni basis states. This clarifies why a
self-adjoint linear operator can be represented by a matrix that, in some basis, is not hermi-tian. A self-adjoint Hamiltonian leads to unitary time-evolution, in the sense that the negative norm is preserved. Given that q, ˙q, ¨q, . . . are self-adjoint, a Hamiltonian which is a generic real function of them is self-adjoint, leading to sensible interacting quantum theory provided that one avoids tachyons, an observation that was previously overlooked. The usual con-dition that the theory should be free of tachyons is generalised to negative norm quantum mechanics.
In section 5 we presented the path-integral formulation of negative-norm quantum me-chanics. The classical Hamiltonian becomes complex. Another new result of this paper is the proof that the normalizable wave functions found in the operator formalism are recovered from the path-integral after performing naive manipulations over ill-defined objects and/or analytic continuations. In particular, the version of the path-integral in Euclidean time tE = it is well defined, and reproduces the usual wave functions taking into account that dq/dtE al-ready coincides with the Dirac-Pauli ˙q.
The fact that (1) our approach leads to normalizable functions and (2) these wave-functions can also be deduced from a well-defined Euclidean path-integral clearly show that the right quantization for ˙q is the Dirac-Pauli one.
Two issues must be addressed before that these results can be used to obtain a predic-tive renormalizable quantum theory of gravity: generalisation to quantum field theory, and generalisation of the Born probabilistic interpretation to negative norms.
Acknowledgments
This work was supported by the ERC grant NEO-NAT, by the Spanish Ministry of Economy and Competitiveness under grant FPA2012-32828, Consolider-CPAN (CSD2007-00042), the grant SEV-2012-0249 of the “Centro de Excelencia Severo Ochoa” Programme and the grant HEPHACOS-S2009/ESP1473 from the C.A. de Madrid. We thank all the colleagues who told us that working with ghosts is equivalent to abandoning physics, and Enrique ´Alvarez, Jos´e R. Espinosa, Antonio Gonz´alez-Arroyo, Martti Raidal, Hardi Veermae, Sergey Sibiryakov, and Guido Altarelli for useful comments and encouragement. This was the content of my last conversation with Guido — the present paper was finalized wondering what Guido would have said.
References
[1] M. Ostrogradski, Mem. Ac. St. Petersbourg VI (1850) 385.
[2] K.S. Stelle, “Renormalization of Higher Derivative Quantum Gravity”, Phys. Rev. D 16 (1977) 953. Recently, the proof of renormalisability was repeated using modern techniques, see G. Rabuffo,“Local cohomology, master equation and renormalization of higher- derivative and nonlocal quantum gravity”, thesis at Pisa University.
[3] E. S. Fradkin and A. A. Tseytlin, “Renormalizable asymptotically free quantum theory of gravity”, Nucl. Phys. B (1982) 201.
[4] I.G. Avramidi and A.O. Barvinsky, “Asymptotic Freedom In Higher Derivative Quantum Gravity”, Phys. Lett. B 159 (1985) 269. I.G. Avramidi, “Covariant methods for the calculation of the effective action in quantum field theory and investigation of higher derivative quantum gravity” [arXiv:hep-th/9510140].
[5] I. Antoniadis and E. T. Tomboulis, “Gauge Invariance and Unitarity in Higher Derivative Quantum Gravity”, Phys. Rev. D 33 (1986) 2756.
[6] E. Elizalde, S.D. Odintsov, A. Romeo, “Renormalization group properties of higher derivative quantum gravity with matter in (4-epsilon)-dimensions”, Nucl. Phys. B462 (1995) 315 [arXiv:hep-th/9502131]. G. Narain, R. Anishetty, “Running Couplings in Quantum Theory of Gravity Coupled with Gauge Fields”, JHEP 1310 (2013) 203 [arXiv:1309.0473].
[7] I. L. Shapiro and A. G. Zheksenaev, “Gauge dependence in higher derivative quantum gravity and the con-formal anomaly problem”, Phys. Lett. B 324 (1994) 286. E.V. Gorbar, I.L. Shapiro, “Renormalization group and decoupling in curved space. 2. The Standard model and beyond”, JHEP 0306 (2003) 004 [ arXiv:hep-ph/0303124]. G. de Berredo-Peixoto, I.L. Shapiro, “Higher derivative quantum gravity with Gauss-Bonnet term”, Phys. Rev. D71 (2004) 064005 [arXiv:hep-th/0412249].
[8] A. Salvio, A. Strumia, “Agravity”, JHEP 1406 (2014) 080 [arXiv:1403.4226].
[9] M.B. Einhorn, D.R.T. Jones, “Naturalness and Dimensional Transmutation in Classically Scale-Invariant Gravity”, JHEP 1503 (2015) 047 [arXiv:1410.8513].
[10] K. Kannike, G. H¨utsi, L. Pizza, A. Racioppi, M. Raidal, A. Salvio, A. Strumia, “Dynamically Induced Planck Scale and Inflation”, JHEP 1505 (2015) 065 [arXiv:1502.01334].
[11] A. Farzinnia, S. Kouwn, “Classically Scale Invariant Inflation, WIMPzilla, and (A)gravity” [arXiv:1512.05890].
[12] R.P. Woodard, “Avoiding dark energy with 1/r modifications of gravity”, Lect. Notes Phys. 720 (2006) 403 [arXiv:astro-ph/0601672]. R.P. Woodard, “The Theorem of Ostrogradsky” [arXiv:1506.02210].
[13] P.A.M. Dirac,“The physical interpretation of quantum mechanics”(1942).
[14] W. Pauli,“On Dirac’s New Method of Field Quantization”, Rev. Mod. Phys. 15 (1943) 175.
[15] W. Heisenberg,“Quantum Theory of Fields and Elementary Particles”, Rev. Mod. Phys. 29 (3) 269. [16] A. Pais and G. E. Uhlenbeck, “On Field theories with nonlocalized action”, Phys. Rev. 79 (1950) 145. [17] T. D. Lee and G. C. Wick, “Negative Metric and the Unitarity of the S Matrix”, Nucl. Phys. B 9 (1969) 209.
T. D. Lee and G. C. Wick, “Finite Theory of Quantum Electrodynamics”, Phys. Rev. D 2 (1970) 1033. [18] R.E. Cutkosky, P. V. Landshoff, D. I. Olive and J. C. Polkinghorne, “A non-analytic S matrix”, Nucl. Phys. B
12 (1969) 281.
[19] S. Coleman, “Acausality”, in “Erice 1969, Ettore Majorana School On Subnuclear Phenomena”, New York (1970) 282.
[20] R.P. Feynman,“Negative probability”in “Quantum implications: Essays in honor of David Bohm”, edited by B.J. Hiley and F.D. Peat (Routledge and Kegan Paul, London, 1987).
[21] D.G. Boulware, D.J. Gross, “Lee-Wick Indefinite Metric Quantization: A Functional Integral Approach”, Nucl. Phys. B 233 (1984) 1.
[22] S.W. Hawking, T. Hertog, “Living with ghosts”, Phys. Rev. D65 (2001) 103515 [arXiv:hep-th/0107088]. See also H. Kleinert, J. Math. Phys. 27 (1986) 12.
[23] G. ’t Hooft, “Unitarity in the Brout-Englert-Higgs Mechanism for Gravity” [arXiv:0708.3184]. G. ’t Hooft, “A class of elementary particle models without any adjustable real parameters”, Found. Phys. 41 (2011) 1829 [arXiv:1104.4543].
[24] B. Grinstein, D. O’Connell, M.B. Wise, “Causality as an emergent macroscopic phenomenon: The Lee-Wick O(N )model”, Phys. Rev. D79 (2008) 105019 [arXiv:0805.2156].
[25] P.D. Mannheim, “Solution to the ghost problem in fourth order derivative theories”, Found. Phys. 37 (2006) 532 [arXiv:hep-th/0608154]. P.D. Mannheim, A. Davidson, “Dirac quantization of the Pais-Uhlenbeck fourth order oscillator”, Phys. Rev. A71 (2004) 042110 [arXiv:hep-th/0408104]. C.M. Bender, “Mak-ing sense of non-Hermitian Hamiltonians”, Rept. Prog. Phys. 70 (2007) 947 [arXiv:hep-th/0703096]. C.M. Bender, P.D. Mannheim, “No-ghost theorem for the fourth-order derivative Pais-Uhlenbeck oscilla-tor model”, Phys. Rev. Lett. 100 (2007) 110402 [arXiv:0706.0207]. C.M. Bender, P.D. Mannheim, “Ex-actly solvable PT-symmetric Hamiltonian having no Hermitian counterpart”, Phys. Rev. D78 (2008) 025022 [arXiv:0804.4190]. C.M. Bender, P.D. Mannheim, “Giving up the ghost”, J. Phys. A41 (2008) 304018 [arXiv:0807.2607]. P.D. Mannheim, “Cosmological Perturbations in Conformal Gravity”, Phys. Rev. D85 (2011) 124008 [arXiv:1109.4119]. P.D. Mannheim, “Comprehensive Solution to the Cosmological Constant, Zero-Point Energy, and Quantum Gravity Problems”, Gen. Rel. Grav. 43 (2009) 703 [arXiv:0909.0212]. P.D. Mannheim, “Making the Case for Conformal Gravity”, Found. Phys. 42 (2011) 388 [arXiv:1101.2186]. P.D. Mannheim, “Advancing the case for P T Symmetry – the Hamiltonian is always P T Symmetric” [arXiv:1506.08432].
[26] A.V. Smilga, “Ghost-free higher-derivative theory”, Phys. Lett. B632 (2005) 433 [arXiv:hep-th/0503213]. A.V. Smilga, “Comments on the dynamics of the Pais-Uhlenbeck oscillator”, SIGMA 5 (2008) 017 [arXiv:0808.0139].
[27] K. Andrzejewski, J. Gonera, P. Maslanka, “Euclidean Path Integral and Higher-Derivative Theories”, Prog. Theor. Phys. 125 (2009) 247 [arXiv:0904.3055]. R. Di Criscienzo, S. Zerbini, “Functional Determinants in Higher Derivative Lagrangian Theories”, J. Math. Phys. 50 (2009) 103517 [arXiv:0907.4265] and Erratum-ibid. 51 (2010) 059901. S.V. Ketov, G. Michiaki, T. Yumibayashi, “Quantizing with a higher time derivative” [arXiv:1110.1155].
[28] T. Biswas, E. Gerwick, T. Koivisto, A. Mazumdar, “Towards singularity and ghost free theories of gravity”, Phys. Rev. Lett. 108 (2012) 031101 [arXiv:1110.5249]
[29] T. Chen, E.A. Lim, “Stabilization of Linear Higher Derivative Gravity with Constraints”, JCAP 1405 (2014) 010 [arXiv:1311.3189].
[30] K. Andrzejewski, J. Gonera, P. Machalski, K. Bolonek-Lason, “On the triviality of higher derivative theories”, Phys. Lett. B706 (2012) 427 [arXiv:1110.0672].
[31] J. Maldacena, “Einstein Gravity from Conformal Gravity” [arXiv:1105.5632].
[32] H. Lu, Y. Pang, C. N. Pope, “Conformal Gravity and Extensions of Critical Gravity”, Phys. Rev. D 84 (2011) 064001 [arXiv:1106.4657].
[33] A. Einstein, B. Podolsky, N. Rosen, “Can Quantum-Mechanical Description of Physical Reality be Considered Complete?”, Phys. Rev. 47 (1935) 777. J. Bell, “On the Einstein-Poldolsky-Rosen paradox”, Physics 1 (1964) 195. A. Aspect, “Bell’s inequality test: more ideal than ever”, Nature 398 (1999) 189.
[34] B. Hasslacher, E. Mottola, Phys. Lett. B 99 (1981) 221.
[35] R. Griffiths, “Consistent histories and the interpretation of quantum mechanics”, J. Stat. Phys 36 (1984) 219. R. Omnes, “Consistent interpretation of quantum mechanics”, Rev. Mod. Phys. 64 (1992) 339. M. Gell-Mann, J.B. Hartle, “Classical equations for quantum systems”, Phys. Rev. D 47 (1993) 3445.
[36] J.G. Cramer, “An Overview of the Transactional Interpretation”, J. Theor. Phys. 27 (1988) 227. J.G. Cramer, “The Transactional Interpretation of Quantum Mechanics and Quantum Nonlocality” [arXiv:1503.00039]. [37] Y. Aharonov, P. G. Bergmann, J. L. Lebowitz, “Time symmetry in the quantum process of measurement”,
